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        <h1 id="chapter-03-集合与关系">chapter 03 集合与关系</h1>
<p>部分作业答案</p>
<p><a href="https:/gitee.com/xingyongkang">邢永康</a></p>
<hr>
<h2 id="p95-4">p95 (4)</h2>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mtext>证明充分条件：</mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>C</mi><mo>⊆</mo><mi>A</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mi>C</mi><mo>∩</mo><mi>A</mi><mo>=</mo><mi>C</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>A</mi><mo>∩</mo><mo stretchy="false">(</mo><mi>B</mi><mo>∪</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mo>∪</mo><mi>C</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mtext>证明必要条件:</mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∀</mi><mi>x</mi><mo>:</mo><mi>x</mi><mo>∈</mo><mi>C</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mi>x</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mo>∪</mo><mi>C</mi><mspace width="1em"/><mtext>(集合并运算的性质)</mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mi>x</mi><mo>∈</mo><mi>A</mi><mo>∩</mo><mo stretchy="false">(</mo><mi>B</mi><mo>∪</mo><mi>C</mi><mo stretchy="false">)</mo><mspace width="1em"/><mtext>（利用集合恒等式）</mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mi>x</mi><mo>∈</mo><mi>A</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo>∴</mo><mi>C</mi><mo>⊆</mo><mi>A</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}
\text{证明充分条件：}\\
C \subseteq A &amp;\Leftrightarrow C \cap A = C \\
A \cap (B\cup C) &amp;=(A\cap B) \cup (A\cap C) \\
&amp;=(A\cap B)\cup C  \\
\text{证明必要条件:}\\
\forall x:  x\in C \\
&amp;\Leftrightarrow x\in (A \cap B) \cup C \quad \text{(集合并运算的性质)}\\
&amp;\Leftrightarrow x \in  A\cap (B\cup C) \quad \text{（利用集合恒等式）}\\
&amp;\Leftrightarrow x \in A \\
\therefore C\subseteq A
\end{align*}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:15em;vertical-align:-7.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:7.75em;"><span style="top:-9.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">证明充分条件：</span></span></span></span><span style="top:-8.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span></span></span><span style="top:-6.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">证明必要条件</span><span class="mord">:</span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:0.59em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:2.09em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:3.59em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel amsrm">∴</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:7.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.25em;"><span style="top:-8.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span style="top:-6.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:1em;"></span><span class="mord text"><span class="mord">(</span><span class="mord cjk_fallback">集合并运算的性质</span><span class="mord">)</span></span></span></span><span style="top:0.59em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:1em;"></span><span class="mord text"><span class="mord cjk_fallback">（利用集合恒等式）</span></span></span></span><span style="top:2.09em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.75em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<hr>
<h2 id="p95-5">p95 (5)</h2>
<h3 id="a">a)</h3>
<p>方法1：</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left right" columnspacing="0em 1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mo lspace="0em" rspace="0em">∵</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mspace width="1em"/><mi>A</mi><mo>−</mo><mi>B</mi><mo>=</mo><mi>A</mi><mo>∧</mo><mi mathvariant="normal">¬</mi><mi>B</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mo lspace="0em" rspace="0em">∴</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mspace width="1em"/><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mi>B</mi><mo stretchy="false">)</mo><mo>−</mo><mi>C</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi mathvariant="normal">¬</mi><mi>B</mi><mo stretchy="false">)</mo><mo>∩</mo><mi mathvariant="normal">¬</mi><mi>C</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>A</mi><mo>∩</mo><mi mathvariant="normal">¬</mi><mi>B</mi><mo>∩</mo><mi mathvariant="normal">¬</mi><mi>C</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>A</mi><mo>∩</mo><mi mathvariant="normal">¬</mi><mo stretchy="false">(</mo><mi>B</mi><mo>∪</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>A</mi><mo>−</mo><mo stretchy="false">(</mo><mi>B</mi><mo>∪</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{align*}
\because &amp; \quad A-B = A \wedge \neg B \\ 
\therefore &amp; \quad (A-B)-C &amp; \\
&amp; =( A \cap \neg B) \cap \neg C \\
&amp; = A \cap \neg B \cap \neg C \\
&amp;= A\cap \neg(B \cup C) \\
&amp;=A-(B\cup C)
\end{align*}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:9em;vertical-align:-4.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.75em;"><span style="top:-6.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel amsrm">∵</span></span></span><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel amsrm">∴</span></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:0.59em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.75em;"><span style="top:-6.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:1em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">¬</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">¬</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">¬</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">¬</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">¬</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">¬</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span><span style="top:0.59em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.25em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-r"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:3.25em;"><span style="top:-5.25em;"><span class="pstrut" style="height:2.84em;"></span><span class="mord"></span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>方法2：</p>
<table>
<thead>
<tr>
<th style="text-align:center">A</th>
<th style="text-align:center">B</th>
<th style="text-align:center">C</th>
<th style="text-align:center">(A-B)-C</th>
<th style="text-align:center">A-(B <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∪</mo></mrow><annotation encoding="application/x-tex">\cup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∪</span></span></span></span> C)</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">0</td>
<td style="text-align:center">0</td>
<td style="text-align:center">0</td>
<td style="text-align:center">0</td>
<td style="text-align:center">0</td>
</tr>
<tr>
<td style="text-align:center">0</td>
<td style="text-align:center">0</td>
<td style="text-align:center">1</td>
<td style="text-align:center">0</td>
<td style="text-align:center">0</td>
</tr>
<tr>
<td style="text-align:center">0</td>
<td style="text-align:center">1</td>
<td style="text-align:center">0</td>
<td style="text-align:center">0</td>
<td style="text-align:center">0</td>
</tr>
<tr>
<td style="text-align:center">0</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">0</td>
<td style="text-align:center">0</td>
</tr>
<tr>
<td style="text-align:center">1</td>
<td style="text-align:center">0</td>
<td style="text-align:center">0</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
</tr>
<tr>
<td style="text-align:center">1</td>
<td style="text-align:center">0</td>
<td style="text-align:center">1</td>
<td style="text-align:center">0</td>
<td style="text-align:center">0</td>
</tr>
<tr>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">0</td>
<td style="text-align:center">0</td>
<td style="text-align:center">0</td>
</tr>
<tr>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">1</td>
<td style="text-align:center">0</td>
<td style="text-align:center">0</td>
</tr>
</tbody>
</table>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mtext>上面的分析</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∀</mi><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>∈</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mi>B</mi><mo stretchy="false">)</mo><mo>−</mo><mi>C</mi><mo stretchy="false">)</mo><mo>↔</mo><mi>x</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mo stretchy="false">(</mo><mi>B</mi><mo>∪</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mi>B</mi><mo stretchy="false">)</mo><mo>−</mo><mi>C</mi><mo>=</mo><mi>A</mi><mo>−</mo><mo stretchy="false">(</mo><mi>B</mi><mo>∪</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}
\text{上面的分析}&amp;\Leftrightarrow 
(\forall x)(x\in ((A-B)-C) \leftrightarrow x \in (A-(B\cup C))  ) \\
&amp; \Leftrightarrow (A-B)-C = A - (B \cup C)
\end{align*}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">上面的分析</span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">((</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)))</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<hr>
<h2 id="p119-4">p119 (4)</h2>
<h3 id="a-1">a)</h3>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∀</mi><mi>y</mi><mo>:</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>y</mi><mo>∈</mo><mi>S</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo>&lt;</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&gt;</mo><mo>∈</mo><mi>S</mi><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mi>x</mi><mo>∈</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇒</mo><mi>y</mi><mo>∈</mo><mi>Y</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>∴</mo><mi>S</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊆</mo><mi>Y</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}
\forall y: &amp; y \in S(A) \\
&amp; \Leftrightarrow (\exist x) ((&lt;x,y&gt;\in S) \wedge(x\in A)) \\
&amp; \Rightarrow y\in Y \\
&amp;\therefore S(A) \subseteq Y
\end{align*}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6em;vertical-align:-2.75em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em;"><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∀</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.75em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em;"><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">∃</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">((</span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mclose">))</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇒</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel amsrm">∴</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.75em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<h3 id="b">b)</h3>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∀</mi><mi>z</mi><mo>:</mo><mo>∈</mo><mo stretchy="false">(</mo><mi>S</mi><mo>∘</mo><mi>T</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∃</mi><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>&lt;</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&gt;</mo><mo>∈</mo><mi>S</mi><mo>∩</mo><mo>&lt;</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo>&gt;</mo><mo>∈</mo><mi>T</mi><mo>∩</mo><mi>x</mi><mo>∈</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∃</mi><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>&lt;</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&gt;</mo><mo>∈</mo><mi>S</mi><mo>∩</mo><mi>x</mi><mo>∈</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∩</mo><mo>&lt;</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo>&gt;</mo><mo>∈</mo><mi>T</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∃</mi><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∩</mo><mo>&lt;</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo>&gt;</mo><mo>∈</mo><mi>T</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mi>z</mi><mo>∈</mo><mi>T</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo>∴</mo><mo stretchy="false">(</mo><mi>S</mi><mo>∘</mo><mi>T</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mi>T</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}
\forall z: \in (S\circ T)(A) \\
&amp;\Leftrightarrow(\exist x)(\exist y)(&lt;x,y&gt;\in S \cap &lt;y,z&gt;\in T \cap x\in A) \\
&amp;\Leftrightarrow(\exist y)((\exist x)(&lt;x,y&gt;\in S \cap x\in A)\cap &lt;y,z&gt;\in T) \\
&amp;\Leftrightarrow(\exist y)(y\in S(A) \cap &lt;y,z&gt;\in T) \\
&amp;\Leftrightarrow z \in T(S(A)\\
\therefore (S\circ T)(A) = T(S(A))



\end{align*}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:9em;vertical-align:-4.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.75em;"><span style="top:-6.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∀</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∘</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:0.59em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel amsrm">∴</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∘</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">))</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em;"><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">∃</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord">∃</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mopen">(</span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord">∩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">∃</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mopen">((</span><span class="mord">∃</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mord">∩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">∃</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mord">∩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mclose">)</span></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.75em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<h3 id="c">c)</h3>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∀</mi><mi>y</mi><mo>:</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>y</mi><mo>∈</mo><mi>s</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>&lt;</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&gt;</mo><mo>∈</mo><mi>S</mi><mo>∧</mo><mi>x</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>&lt;</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&gt;</mo><mo>∈</mo><mi>S</mi><mo>∧</mo><mo stretchy="false">(</mo><mi>x</mi><mo>∈</mo><mi>A</mi><mo>∪</mo><mi>x</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo>&lt;</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&gt;</mo><mo>∈</mo><mi>S</mi><mo>∧</mo><mi>x</mi><mo>∈</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mo>&lt;</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&gt;</mo><mo>∈</mo><mi>S</mi><mo>∧</mo><mi>x</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>&lt;</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&gt;</mo><mo>∈</mo><mi>S</mi><mo>∧</mo><mi>x</mi><mo>∈</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>&lt;</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&gt;</mo><mo>∈</mo><mi>S</mi><mo>∧</mo><mi>x</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∪</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo>∴</mo><mi>S</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>S</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∪</mo><mi>S</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}
\forall y: &amp;y\in s(A\cup B) \\
&amp;\Leftrightarrow (\exist x)(&lt;x,y&gt;\in S \wedge x\in (A\cup B))\\
&amp;\Leftrightarrow (\exist x)(&lt;x,y&gt;\in S \wedge (x\in A \cup x\in B)) \\
&amp;\Leftrightarrow (\exist x)((&lt;x,y&gt;\in S \wedge x\in A)\cup(&lt;x,y&gt;\in S \wedge x\in B))\\
&amp;\Leftrightarrow (\exist x)(&lt;x,y&gt;\in S \wedge x\in A)\cup(\exist x)(&lt;x,y&gt;\in S \wedge x\in B)\\
&amp;\Leftrightarrow y\in S(A) \cup y \in S(B) \\
\therefore S(A\cup B) = S(A)\cup S(B)
\end{align*}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:10.5em;vertical-align:-5em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.5em;"><span style="top:-7.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∀</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span></span></span><span style="top:-6.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-0.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:1.34em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel amsrm">∴</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.5em;"><span style="top:-7.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">s</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span><span style="top:-6.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">∃</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">))</span></span></span><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">∃</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">))</span></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">∃</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">((</span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">))</span></span></span><span style="top:-1.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">∃</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord">∃</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span><span style="top:-0.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.5em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<h3 id="d">d)</h3>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∀</mi><mi>y</mi><mo>:</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>y</mi><mo>∈</mo><mi>s</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>&lt;</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&gt;</mo><mo>∈</mo><mi>S</mi><mo>∧</mo><mi>x</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>&lt;</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&gt;</mo><mo>∈</mo><mi>S</mi><mo>∧</mo><mo stretchy="false">(</mo><mi>x</mi><mo>∈</mo><mi>A</mi><mo>∩</mo><mi>x</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>&lt;</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&gt;</mo><mo>∈</mo><mi>S</mi><mo>∧</mo><mi>x</mi><mo>∈</mo><mi>A</mi><mo>∧</mo><mi>x</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo>&lt;</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&gt;</mo><mo>∈</mo><mi>S</mi><mo>∧</mo><mi>x</mi><mo>∈</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mo>&lt;</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&gt;</mo><mo>∈</mo><mi>S</mi><mo>∧</mo><mi>x</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇒</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>&lt;</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&gt;</mo><mo>∈</mo><mi>S</mi><mo>∧</mo><mi>x</mi><mo>∈</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∩</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>&lt;</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&gt;</mo><mo>∈</mo><mi>S</mi><mo>∧</mo><mi>x</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇔</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo>∴</mo><mi>S</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo stretchy="false">)</mo><mo>⊆</mo><mi>S</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∪</mo><mi>S</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}
\forall y: &amp;y\in s(A\cap B) \\
&amp;\Leftrightarrow (\exist x)(&lt;x,y&gt;\in S \wedge x\in (A\cap B))\\
&amp;\Leftrightarrow (\exist x)(&lt;x,y&gt;\in S \wedge (x\in A \cap x\in B)) \\
&amp;\Leftrightarrow (\exist x)(&lt;x,y&gt;\in S \wedge x\in A\wedge x\in B)\\
&amp;\Leftrightarrow (\exist x)((&lt;x,y&gt;\in S \wedge x\in A)\wedge(&lt;x,y&gt;\in S \wedge x\in B))\\
&amp;\Rightarrow (\exist x)(&lt;x,y&gt;\in S \wedge x\in A)\cap(\exist x)(&lt;x,y&gt;\in S \wedge x\in B)\\
&amp;\Leftrightarrow y\in S(A) \cap y \in S(B) \\
\therefore S(A\cup B) \subseteq S(A)\cup S(B)
\end{align*}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:12em;vertical-align:-5.75em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.25em;"><span style="top:-8.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∀</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span></span></span><span style="top:-6.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:0.59em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:2.09em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel amsrm">∴</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.75em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.25em;"><span style="top:-8.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">s</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span><span style="top:-6.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">∃</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">))</span></span></span><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">∃</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">))</span></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">∃</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">∃</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">((</span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">))</span></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇒</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">∃</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord">∃</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span><span style="top:0.59em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>

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